What are some conflicts in the play?

There are several conflicts worth mentioning in this play. First, Professor Higgins and Colonel Pickering have a playful "bet" that Higgins can't make a common flower girl speak and act like a duchess.  There is the issue of language, pronunciation, and what we actually hear and say as opposed to what we think we hear and say.

There is the issue that Eliza Doolittle does want to improve her situation in life, but is rather stuck in her position as flower girl since her income fluctuates and her education is limited.

There is the issue of what to do with Eliza once she learns to speak and act properly.  She can't go back to selling flowers, yet she has no real place among the aristocratic society she has been trained to infiltrate.

There is the conflict with Higgins and everyone else--including his mother--since he considerably rude, late, and isn't really the ideal role model for Eliza's "proper" education.

There is also the minor conflict of Eliza's father, who has come into some money himself and is struggling about his impending marriage and adjusting to suddenly becoming "a gentleman".

There is also the conflict of Eliza's romantic feelings for Higgins and for Freddy whom she meets at one of the outings to test her ability to fool others into thinking that she is high society.